Also you need to transfer this moving/stopping and standing still to other areas. When you halter him. Make him stand still before you lead him. When leading him, make him stand still, and move at differing paces and stand still when you tell him to. And don't talk to him or pet him.
He should be standing there, just waiting to see what you are going to do, not fidgeting, nor being praised.
To me, your comment about girth being tight enough? You may be overgirthing. Too tight a girth is painful, and also may have metal where it is irritating horse or compromising blood supply too. For that matter? You can mount without any girth at all. Not recommended but shows that girth doesn't need to be giving horse wasp waist.
I always shorten offside rein, so that if horse does move? Hindquarters come towards me, so I can settle in saddle.
But make horse stand quietly under you at differing times. Not always constant movement, and don't just let horse move off after a couple of seconds. Minutes can go by and horse needs to stand there. Period. Again, no praising or petting horse.
And making sure you aren't jabbing with toe is good advice, and also making sure you haven't gotten his mane trapped under saddle/pad, as that pulls and will make horse move. Pad/blanket should be tented under gullet of saddle to avoid this.
Horses make me a better person.